PR (complexity)

PR is the complexity class of all primitive recursive functions—or, equivalently, the set of all formal languages that can be decided in time bounded by such a function. This includes addition, multiplication, exponentiation, tetration, etc.

The Ackermann function is an example of a function that is not primitive recursive, showing that PR is strictly contained in R (Cooper 2004:88).

On the other hand, we can "enumerate" any recursively enumerable set (see also its complexity class RE) by a primitive-recursive function in the following sense: given an input $(M, k)$ , where $M$ is a Turing machine and $k$ is an integer, if $M$ halts within $k$ steps then output $M$ ; otherwise output nothing. Then the union of the outputs, over all possible inputs ( $M$ , $k$ ), is exactly the set of $M$ that halt.

PR strictly contains ELEMENTARY.

PR does not contain "PR-complete" problems (assuming, e.g., reductions that belong to ELEMENTARY).

Hierarchy

The PR class can be divided into an infinite hierarchy of increasingly large complexity levels, according to the fast-growing hierarchy.

The $\text{𝐅}_0$ class is the class of problems that can be solved in $n + O(1)$ time. That is, there exists a Turing machine and a constant $C$ , such that given an input of length $n$ , the machine solves it and halts within $n + C$ steps.

The $\text{𝐅}_1$ class is the class of problems that can be solved in $\mathsf{poly}(n)$ time.

The $\text{𝐅}_2$ class is ELEMENTARY.

The $\text{𝐅}_3$ class is TOWER, which can be equivalently written as the class of problems that can be solved in tetration-time.

The union $\bigcup_{n \in \N} \text{𝐅}_n$ is PR.

In practice, many problems that are not in PR but just beyond it are $\text{𝐅}_\omega$ -complete (Schmitz 2016).

References

{{cite book|author=S. Barry Cooper |year=2004|title=Computability Theory|publisher= Chapman & Hall|isbn=1-58488-237-9}}
{{cite book|author=Herbert Enderton |year=2011|title=Computability Theory|publisher= Academic Press|isbn=978-0-12-384-958-8}}
{{cite journal|arxiv=1312.5686|doi=10.1145/2858784|title=Complexity Hierarchies beyond Elementary|year=2016|last1=Schmitz|first1=Sylvain|journal=ACM Transactions on Computation Theory|volume=8|pages=1–36|s2cid=15155865 }}

External links

{{ComplexityZoo|PR|P#pr}}.

Category:Complexity classes